Abstract
Steady two-dimensional turbulent free-surface flow in a channel with a slightly uneven bottom is considered. The shape of the unevenness of the bottom can be in the form of a bump or a ramp of very small height. The slope of the channel bottom is assumed to be small, and the bottom roughness is assumed to be constant. Asymptotic expansions for very large Reynolds numbers and Froude numbers close to the critical value \({Fr} = 1\), respectively, are performed. The relative order of magnitude of two small parameters, i.e. the bottom slope and \(({Fr}-1)\), is defined such that no turbulence modelling is required. The result is a steady-state version of an extended Korteweg–de Vries equation for the surface elevation. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. An exact solution describing stationary solitary waves of the classical shape is obtained for a bottom of a particular shape. For more general shapes of ramps and bumps, stationary solitary waves of the classical shape are also obtained as a first approximation in the limit of small, but nonzero, dissipation. With the exception of an eigensolution for a ramp, an outer region has to be introduced. The outer solution describes a ’tail’ that is attached to the stationary solitary wave. In addition to the solutions of the solitary-wave type, solutions of smaller amplitudes are obtained both numerically and analytically. Experiments in a water channel confirm the existence of both types of stationary single waves.
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Binder, B.J., Blyth, M.G., Balasuriya, S.: Non-uniqueness of steady free-surface flow at critical Froude number. EPL 105(4), 44003 (2014). https://doi.org/10.1209/0295-5075/105/44003
Binder, B.J., Dias, F., Vanden-Broeck, J.M.: Influence of rapid changes in a channel bottom on free-surface flows. IMA J. Appl. Math. 73, 254–273 (2008). https://doi.org/10.1093/imamat/hxm049
Bowen, M.K., Smith, R.: Derivative formulae and errors for non-uniformly spaced points. Proc. R. Soc. Lond. Ser. A 461(2059), 1975–1997 (2005). https://doi.org/10.1098/rspa.2004.1430
Camassa, R., Wu, Y.T.: Stability of forced steady solitary waves. Philos. Trans. R. Soc. Lond. A 337(1648), 429–466 (1991). https://doi.org/10.1098/rsta.1991.0133. http://rsta.royalsocietypublishing.org/content/337/1648/429
Camassa, R., Wu, Y.T.: Stability of some stationary solutions for the forced KdV equation. Physica D 51(1–3), 295–307 (1991). https://doi.org/10.1016/0167-2789(91)90240-A. http://www.sciencedirect.com/science/article/pii/016727899190240A
Cantero-Chinchilla, F.N., Castro-Orgaz, O., Khan, A.A.: Depth-integrated nonhydrostatic free-surface flow modelling using weighted-averaged equations. Int. J. Numer. Methods Fluids 87, 27–50 (2018). https://doi.org/10.1002/fld.4481
Castro-Orgaz, O.: Weakly undular hydraulic jump: effects of friction. J. Hydr. Res. 48(4), 453–465 (2010). https://doi.org/10.1080/00221686.2010.491646
Castro-Orgaz, O., Chanson, H.: Near-critical free-surface flows: real fluid flow analysis. Environ. Fluid Mech 11(5), 499–516 (2011)
Castro-Orgaz, O., Hager, W.H.: Observations on undular hydraulic jump in movable bed. J. Hydr. Res. 49(5), 689–692 (2011)
Castro-Orgaz, O., Hager, W.H., Dey, S.: Depth-averaged model for undular hydraulic jump. J. Hydr. Res. 53(3), 351–363 (2015)
Castro-Orgaz, O., Hager, W.H.: Non-hydrostatic Free Surface Flows. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin (2017). https://doi.org/10.1007/978-3-319-47971-2
Chanson, H.: The Hydraulics of Open Channel Flow: An Introduction, 2nd edn. Elsevier, Butterworth-Heinemann, Oxford (2004). http://www.ebook.de/de/product/2867163/hubert_the_university_of_queensland_australia_chanson_hydraulics_of_open_channel_flow.html
Chardard, F., Dias, F., Nguyen, H.Y., Vanden-Broeck, J.M.: Stability of some stationary solutions to the forced KdV equation with one or two bumps. J. Eng. Math. 70(1–3), 175–189 (2011). https://doi.org/10.1007/s10665-010-9424-6
Chow, V.T.: Open-Channel Hydraulics. McGraw-Hill, New York (1959)
Christov, C.I., Velarde, M.: Dissipative solitons. Physica D 86(1–2), 323–347 (1995). https://doi.org/10.1016/0167-2789(95)00111-G. http://www.sciencedirect.com/science/article/pii/016727899500111G
Dias, F., Vanden-Broeck, J.M.: Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170 (1989). https://doi.org/10.1017/S0022112089002260. https://www.cambridge.org/core/article/div-class-title-open-channel-flows-with-submerged-obstructions-div/D1418B1CFA3C476168345279041C2197
Dias, F., Vanden-Broeck, J.M.: Generalised critical free-surface flows. J. Eng. Math. 42(3), 291–301 (2002). https://doi.org/10.1023/A:1016111415763
Dutykh, D.: Visco-potential free-surface flows and long wave modelling. Eur. J. Mech. B Fluids 28(3), 430–443 (2009)
Dutykh, D., Dias, F.: Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Acad. Sci. Paris Ser. I 345(2), 113–118 (2007)
Forbes, L.K., Schwartz, L.W.: Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314 (1982). https://doi.org/10.1017/S0022112082000160. https://www.cambridge.org/core/article/div-class-title-free-surface-flow-over-a-semicircular-obstruction-div/B64B4833893C39057B2E1479E872FBEA
Gersten, K.: Turbulent boundary layers I: fundamentals. In: Kluwick, A. (ed.) Recent Advances in Boundary Layer Theory, CISM Courses and Lectures, vol. 390, pp. 107–144. Springer, Wien (1998). https://doi.org/10.1007/978-3-7091-2518-2_5
Gong, L., Shen, S.S.: Multiple supercritical solitary wave solutions of the stationary forced Korteweg–de Vries equation and their stability. SIAM J. Appl. Math. 54(5), 1268–1290 (1994)
Gotoh, H., Yasuda, Y., Ohtsu, I.: Effect of channel slope on flow characteristics of undular hydraulic jumps. In: Brebbia, C., do Carmo, J.S.A. (eds.) River Basin Management III, vol. 83, pp. 33–42. WIT Press, Southampton (2005)
Grillhofer, W.: Der wellige Wassersprung in einer turbulenten Kanalströmung mit freier Oberfläche. Dissertation. Technische Universität Wien, Vienna (2002)
Grillhofer, W., Schneider, W.: The undular hydraulic jump in turbulent open channel flow at large Reynolds numbers. Phys. Fluids 15(3), 730–735 (2003)
Grimshaw, R.: Exponential asymptotics and generalized solitary waves. In: Steinrück H. (ed.) Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances, CISM Courses and Lectures, vol. 523, pp. 71–120. Springer, Wien, New York (2010)
Grimshaw, R.: Transcritical flow past an obstacle. ANZIAM J. 52(1), 2–26 (2010)
Grimshaw, R., Zhang, D.H., Chow, K.W.: Generation of solitary waves by transcritical flow over a step. J. Fluid Mech. 587, 235–254 (2007). https://doi.org/10.1017/S0022112007007355
Hager, W.H., Castro-Orgaz, O.: Transcritical flow in open channel hydraulics: from Böss to De Marchi. J. Hydr. Eng. 142(1), 02515003 (2016). https://doi.org/10.1061/(ASCE)HY.1943-7900.0001091
Handler, R.A., Swean Jr., T.F., Leighton, R.I., Swearingen, J.D.: Length scales and the energy balance for turbulence near a free surface. AIAA J. 31(11), 1998–2007 (1993)
Hassanzadeh, R., Sahin, B., Ozgoren, M.: Large eddy simulation of free-surface effects on the wake structures downstream of a spherical body. Ocean Eng. 54, 213–222 (2012)
Hös, C., Kullmann, L.: A numerical study on the free-surface channel flow over a bottom obstacle. In: Conference on Modelling Fluid Flow (CMFF’06). The 13th International Conference on Fluid Flow Technologies, pp. 500–506. Budapest, Hungary (2006)
Jurisits, R.: Wellige Wassersprünge bei nicht voll ausgebildeter turbulenter Zuströmung. Dissertation, Technische Universität Wien, Vienna (2012)
Jurisits, R.: Transient numerical solutions of an extended Korteweg–de Vries equation describing solitary waves in open-channel flow. Period. Polytech. Mech. Eng. 61(1), 55–59 (2017)
Jurisits, R., Schneider, W.: Undular hydraulic jumps arising in non-developed turbulent flows. Acta Mech. 223(8), 1723–1738 (2012). https://doi.org/10.1007/s00707-012-0666-4
Jurisits, R., Schneider, W., Bae, Y.S.: A multiple-scales solution of the undular hydraulic jump problem. Proc. Appl. Math. Mech. (PAMM) 7(1), 4120,007–4120,008 (2007). https://doi.org/10.1002/pamm.200700755
Kichenassamy, S., Olver, P.J.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23(5), 1141–1166 (1992)
Kluwick, A.: Interacting laminar and turbulent boundary layers. In: Kluwick A. (ed.) Recent Advances in Boundary Layer Theory, CISM Courses and Lectures, vol. 390, pp. 231–330. Springer, Wien (1998)
Knickerbocker, C.J., Newell, A.C.: Shelves and the Korteweg–de Vries equation. J. Fluid Mech. 98, 803–818 (1980). https://doi.org/10.1017/S0022112080000407. http://journals.cambridge.org/article_S0022112080000407
Komori, S., Nagaosa, R., Murakami, Y., Chiba, S., Ishii, K., Kuwahara, K.: Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas-liquid interface. Phys. Fluids A 5(1), 115–125 (1993)
Lennon, J.M., Hill, D.F.: Particle image velocity measurements of undular and hydraulic jumps. J. Hydr. Eng. 132(12), 1283–1294 (2006). https://doi.org/10.1061/(ASCE)0733-9429(2006)132:12(1283)
Lovecchio, S., Zonta, F., Soldati, A.: Upscale energy transfer and flow topology in free-surface turbulence. Phys. Rev. E 91, 033,010 (2015). https://doi.org/10.1103/PhysRevE.91.033010
Madsen, P.A., Svendsen, I.A.: On the form of the integrated conservation equations for waves in the surf zone. Prog. Rep. 48, 31–39 (1979). (Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark)
Marchant, T.R.: Coupled Korteweg–de Vries equations describing, to high-order, resonant flow of a fluid over topography. Phys. Fluids 11(7), 1797–1804 (1999). https://doi.org/10.1063/1.870044
Miles, J.W.: Solitary wave evolution over a gradual slope with turbulent friction. J. Phys. Oceanogr. 13(3), 551–553 (1983). https://doi.org/10.1175/1520-0485(1983)013<0551:SWEOAG>2.0.CO;2
Miles, J.W.: Wave evolution over a gradual slope with turbulent friction. J. Fluid Mech. 133, 207–216 (1983). https://doi.org/10.1017/S002211208300186X
Müllner, M.: Solutions of an extended KdV equation describing single stationary waves with strong or weak downstream decay in turbulent open-channel flow. ZAMM Z. Angew. Math. Mech 98(1), 7–30 (2018). https://doi.org/10.1002/zamm.201700040
Müllner, M., Schneider, W.: Asymptotic solutions of an extended Korteweg–de Vries equation describing solitary waves with weak or strong downstream decay in turbulent open-channel flow. Proc. Appl. Math. Mech. (PAMM) 15(1), 491–492 (2015). https://doi.org/10.1002/pamm.201510236
Müllner, M., Schneider, W.: Stationary single waves in turbulent open-channel flow. Proc. Appl. Math. Mech. (PAMM) 17, 683–684 (2017). https://doi.org/10.1002/pamm.201710310
Narayanan, C., Lakehal, D., Botto, L., Soldati, A.: Mechanisms of particle deposition in a fully developed turbulent open channel flow. Phys. Fluids 15(3), 763–775 (2003)
Newell, A.C.: Solitons in Mathematics and Physics. No. 48 in CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1985). http://gen.lib.rus.ec/book/index.php?md5=c8b76e22856ecc29045c928ee4541e12
Nezu, I., Rodi, W.: Open-channel flow measurements with a laser Doppler anemometer. J. Hydr. Eng. 112(5), 335–355 (1986). https://doi.org/10.1061/(ASCE)0733-9429(1986)112:5(335)
Ohtsu, I., Yasuda, Y., Gotoh, H.: Hydraulic condition for undular-jump formations. J. Hydr. Res. 39(2), 203–209 (2001)
Ohtsu, I., Yasuda, Y., Gotoh, H.: Flow conditions of undular hydraulic jumps in horizontal rectangular channels. J. Hydr. Eng. 129(12), 948–955 (2003)
Pelinovsky, E.N., Stepanyants, Y., Talipova, T.: Nonlinear dispersion model of sea waves in the coastal zone. J. Korean Soc. Coast. Ocean Eng. 5(4), 307–317 (1993)
Rednikov, A.Y., Velarde, M.G., Ryazantsev, Y.S., Nepomnyashchy, A.A., Kurdyumov, V.N.: Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation. Acta Appl. Math. 39(1–3), 457–475 (1995)
Rodi, W.: Turbulence Models and their Application in Hydraulics, 3rd edn. Balkema, Rotterdam (1993)
Rostami, F., Yazdi, S.R.S., Said, M.A.M., Shahrokhi, M.: Numerical simulation of undular jumps on graveled bed using volume of fluid method. Water Sci. Technol. 66(5), 909–917 (2012)
Schlichting, H., Gersten, K.: Boundary-Layer Theory, 9th edn. Springer, Berlin (2017)
Schneider, W.: Solitary waves in turbulent open-channel flow. J. Fluid Mech. 726, 137–159 (2013)
Schneider, W., Jurisits, R., Bae, Y.S.: An asymptotic iteration method for the numerical analysis of near-critical free-surface flows. Int. J. Heat Fluid Flow 31(6), 1119–1124 (2010). https://doi.org/10.1016/j.ijheatfluidflow.2010.07.004. http://www.sciencedirect.com/science/article/pii/S0142727X1000130X
Schneider, W., Yasuda, Y.: Stationary solitary waves in turbulent open-channel flow: analysis and experimental verification. J. Hydr. Eng. 142(1), 04015,035 (2015). https://doi.org/10.1061/(ASCE)HY.1943-7900.0001056. http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0001056
Scott, A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford Texts in Applied and Engineering Mathematics, 2nd edn. Oxford University Press, Oxford (2003)
Steinrück, H., Schneider, W., Grillhofer, W.: A multiple scales analysis of the undular hydraulic jump in turbulent open channel flow. Fluid Dyn. Res. 33(1–2), 41–55 (2003). https://doi.org/10.1016/S0169-5983(03)00041-8. http://www.sciencedirect.com/science/article/pii/S0169598303000418. In memoriam: Prof. Philip Gerald Drazin 1934-2002
Svendsen, I.A., Veeramony, J., Bakunin, J., Kirby, J.T.: The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57 (2000)
Sykes, R.: An asymptotic theory of incompressible turbulent boundary-layer flow over a small hump. J. Fluid Mech. 101, 647–670 (1980)
Vanden-Broeck, J.M.: Free-surface flow over an obstruction in a channel. Phys. Fluids 30(8), 2315–2317 (1987). https://doi.org/10.1063/1.866121. http://aip.scitation.org/doi/abs/10.1063/1.866121
Acknowledgements
Open access funding provided by TU Wien (TUW). The authors are indebted to Prof. Oscar Castro-Orgaz for his encouragement to include variations in bottom shape into the asymptotic analysis of turbulent free-surface flow. Dr. Richard Jurisits’ yet unpublished numerical solutions of the extended KdV equation helped the authors to cope with numerical problems and find the solutions of the second kind. Dr. Christoph Buchner, Dr. Richard Jurisits, Dr. Bernhard Scheichl and a reviewer provided useful references. The reviewers’ comments led to various improvements of the presentation, including the supply of additional information in three appendices. Mr. Dominik Murschenhofer prepared the file. Finally, financial support by Androsch International Management Consulting GmbH is gratefully acknowledged.
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Schneider, W., Müllner, M. & Yasuda, Y. Near-critical turbulent open-channel flows over bumps and ramps. Acta Mech 229, 4701–4725 (2018). https://doi.org/10.1007/s00707-018-2230-3
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DOI: https://doi.org/10.1007/s00707-018-2230-3